Ind. Eng. Chem. Res. 2005, 44, 6095-6102
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Scale-up Methods for Fast Competitive Chemical Reactions in Pipeline Mixers Rupert Anthony (Tony) Taylor† Emergent Technologies, 2508 Ashley Worth Boulevard, Austin, Texas 78734
W. Roy Penney* Chemical Engineering Department, University of Arkansas, Fayetteville, Arkansas 72701
Hanh X. Vo‡ Dow Corning Corporation, P.O. Box 0995, Mail No. 128, Midland, Michigan 48686-0995
An experimental study investigated the scale-up methods for fast competitive/consecutive reactions in static mixers. Twelve-element Kenics helical element mixers of 1/8-, 1/4-, and 1/2-in.diameters were tested using the fourth Bourne reaction system as a test reaction. The fourth Bourne reaction is a parallel reaction system where the acid-catalyzed hydrolysis of 2,2dimethoxypropane (DMP) is conducted in parallel with the neutralization of HCl with NaOH. With very rapid mixing, the HCl catalyst is neutralized by NaOH and minimal hydrolysis of DMP occurs. The main feed contained 200 mol/m3 of DMP and 210 mol/m3 of NaOH (5% stoichometric excess relative to HCl in the side stream). The side stream, which was introduced through two radial cylindrical feed ports at the midpoint of the third mixer element, contained 2000 mol/m3 of HCl. The two streams were combined in the mixer at a volumetric flow ratio of 10:1, main to side. By analysis of the reactor effluent for DMP hydrolysis products of acetone and methanol, the effectiveness of the mixer-reactor in promoting the acid-base reaction was quantified. The parameters used to evaluate the static mixer performance at different sizes are (1) the Reynolds number (experiment Re varied from 550 to 17 000), (2) the turbulent energy dissipation rate, and (3) the residence time in the mixer. At low flow rates and low energy dissipation, micromixing controlled and equal power dissipation was an acceptable scale-up criterion. At high flow rates and high energy dissipation, mesomixing controlled and equal residence time was the proper scale-up criterion. The recommended scale-up procedure is as follows: (1) Maintain geometrical similarity. (2) Maintain the following parameters constant: (A) ratio of the main to side feed rates, (B) ratio of the main to side feed velocities, (C) ratio of the reagent concentration in the main and side feeds, (D) number of mixing elements in the mixer, and (E) side feed orientation and location relative to the mixing elements. (3) Analyze the experimental data to determine the controlling mixing mechanism, either micromixing or mesomixing: (A) for micromixing control, scale-up using equal power dissipation; (B) for mesomixing control, scale-up using equal mixer residence time. 1. Introduction In reactor design, the fluid mixing rate can have an important effect on the yield and selectivity of chemical reactions. For slow reactions, the rate of mixing is not a controlling step; therefore, the mixing rate is not important in determining the yield or selectivity. However, for reactions with fast kinetics and especially for fast competitive/consecutive (C/C) reactions, slow mixing rates can limit the overall reaction rate and/or promote slow side reactions relative to fast desired reactions. Thus, for many industrially important fast competitive reactions, slow mixing lowers the yield of desired product(s). Acid-base neutralization in the presence of organic substrates is the most commonly encountered example where poor mixing can promote undesired side reactions. Neutralization is the desired reaction; however,
many organic species are very reactive under high concentrations of acid or base. Rapid mixing will promote the very fast neutralization reaction, whereas slow mixing will allow organic species, in the presence of acids or bases, to react by substitution or decomposition, thereby producing side products. Fast C/C reaction systems are particularly prevalent in the pharmaceutical and specialty chemical industries; Paul1 has given several examples of fast C/C chemical reaction systems encountered in the pharmaceutical industry. Fast C/C reactions are conducted in agitated vessels,1-12 agitated vessels with recycle loops,13-15 and continuous-flow static mixers.16-19 Static mixers are commercially available in several configurations. This study used a 12-element Kenics helical element mixer (HEM), which is depicted in Figure 1. 2. Literature Review
* To whom correspondence should be addressed. Tel.: (501) 575-5681. † Tel.: (512) 263-3232. ‡ Tel.: (517) 496-5876.
2.1. Multiple Fast Reactions. For single-phase systems, Knight13 indicates that reactions are typically considered fast if they have half-lives of 10-3-10-6 s.
10.1021/ie040237u CCC: $30.25 © 2005 American Chemical Society Published on Web 04/05/2005
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Figure 1. Kenics HEM static mixer.
C/C fast reactions are a convenient experimental tool for studying the effect of mixing on chemical reactions. With a C/C fast reaction system, the reactor effluent composition is indicative of the mixing effectiveness in promoting desired reactions. Baldyga and Bourne2 [Chapter 10, p 643] have developed at least four different fast competitive and/ or C/C reaction systems for studying the effect of mixing on reactor yields. The first three Bourne reactions are documented by Baldyga and Bourne2 [Chapter 10, pp 642-669] and Paul et al.9 [Chapter 13, p 786]. The first three Bourne reactions are mentioned here only in terms of their nature and relative reaction rates. The first Bourne reaction is a C/C reaction system of 1-naphthol reacting with diazotized sulfanic acid to produce o and p isomers. It is relatively slow and is only suitable for studying fast C/C reactions in agitated vessels. The second Bourne reaction is an extension of the first Bourne reaction, in which 1-naphthol and 2-naphthol react competitively with diazotized sulfanic acid as the initial steps in the reaction scheme. The second Bourne reaction is about 5 times faster than the first Bourne reaction; consequently, it is suitable for use with pipeline mixer reactors. The third Bourne reaction is the hydrolysis of ethyl chloroacetate (3a) or methyl chloracetate (3b) reacting with NaOH in competition with the parallel reaction of NaOH with HCl. Both 3a and 3b are suitable for use in agitated vessels but are too slow to be used in pipeline mixers. The fourth Bourne reaction was used in this study. It consists of HCl reacting with NaOH as the fast reaction and the acid-catalyzed decomposition of 2,2dimethoxypropane (DMP) for the slow reaction. It has been explained by Walker20 and Baldyga et al.21 Details of the reaction system are
A + B w P1 + P2 [k ) 1.3 × 1011 m3/kg‚mol‚s at 25 °C] (1) C + P2 w Q1 + 2Q2 [k ) 700 m3/kg‚mol‚s at 25 °C and CNaCl ) 100 gmol/m3] (2) where A ) sodium hydroxide (NaOH), B ) hydrochloric acid (HCl), C ) dimethoxypropane (CH3C{OCH3}2CH3; i.e., DMP), P1 ) sodium chloride (NaCl), P2 ) water (H2O), Q1 ) acetone (CH3COCH3), Q2 ) methanol (CH3OH), and solvent ) 25 wt % ethanol aqueous solution. The hydrolysis reaction of DMP (eq 2) is catalyzed by HCl, and its kinetics is given by Walker20 and Baldyga et al.21
rC ) -kCCCB
(3)
where CB ) H+ concentration and CC ) DMP concentration. For the fourth Bourne reaction, with equal starting concentrations of reagents, the initial rate of the slow reaction, the hydrolysis of DMP, is about 20 times greater than the hydrolysis of ethyl chloracetate, which is the slow reaction in the (3a) Bourne reaction. For the fourth Bourne reaction, the yield of the slow reaction products, acetone and methanol, is indicative of the level of mixing. If instantaneous mixing is achieved, none of DMP will be reacted. 2.2. Design and Scale-up Correlations. The literature concerning the design and scale-up for fast C/C reactions can be categorized as follows: for agitated vessels, Baldyga and Bourne,2 Bourne and Yu,3 Drain,6 Fasano and Penney,7 Nienow et al.,8 Tipnis,10 Tipnis et al.,11 and Yu;12 for agitated vessels with recycle loops, Knight,13 and Knight et al.;14 for pipeline mixers, Bourne et al.,16 Hearn,17 Baldyga et al.,18 and Taylor.19 The important findings concerning scale-up are summarized as follows. Paul1 found that equal blend time (i.e., equal impeller speed for geometrically similar reactors) was the appropriate scale-up criterion for the first Bourne reaction, with constant feed time, in reactors agitated with three retreat-blade glass-coated impellers. Bourne and Yu3,12 found that scale-up in agitated vessels can be accomplished at equal impeller power per unit volume (or mass) of batch by increasing the feed time of the continuously fed reagent as the batch size increased. Tipnis10 and Tipnis et al.11 determined, for the third (3a) Bourne reaction, that equal blend time was the proper scale-up criterion for a constant feed time of the fed reagent. Drain et al.6,8 also found that equal blend time was the proper scale-up criterion, for a constant feed time, using a relatively slow C/C reaction scheme. Knight13 and Knight et al.14 concluded that an agitated vessel/recycle loop/static mixer system could be scaled up using the following criteria: (A) at least equal agitator power per unit volume in the agitated vessel, (B) equal to or greater than feed-pipe jet velocity, (C) constant residence time in the static mixer, (D) equal or lower coefficient of variation (COV) at the exit of the mixer, and (E) initial molar feed ratio of the fed reagent to recycled batch reagent > 2.5. The literature concerning the design and scale-up for fast C/C reactions in pipeline mixers is rather modest compared to the extensive literature for agitated vessels. The important findings concerning scale-up of pipeline mixers used for fast C/C reactions are summarized by Baldyga and Bourne,2 Bourne et al.,16 Hearn,17 and Taylor.19 Hearn17 did an extensive study of the velocity and turbulence profiles and fast reaction performance in four types of static mixers (including the helical element Kenics HEM), and he investigated, both experimentally and theoretically, scale-up procedures. Hearn17 defined three “motionless mixer time scales”: (1) micromixing time scale, (2) mesomixing time scale, and (3) macromixing time scale. The largest time scale dominates scale-up. Hearn determined the following scale-up criteria: micromixing
mesomixing
macromixing
equal power dissipation
equal resonance time
equal residence time
Baldyga and Bourne2 called the time scales “time constants” [p 762], and they used the following defini-
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tions for the “time constants”: (1) micromixing, τE; (2) mesomixing: turbulent dispersion, τD; (3) mesomixing: turbulent disintegration of the entering feed, τS0; (4) mesomixing: turbulent disintegration of the relaxed feed, τSr. Baldyga and Bourne’s2 [pp 762-764] analysis will be used here, and the appropriate time scales will be calculated per their “Worked Example 12.1: Time Constants for a Kenics Mixer”. The largest time scale indicates the controlling mixing mechanism.
τE ) 1/E ) 17.3(ν/)1/2 [Baldyga and Bourne,2 eq 12.5, pp 733 and 763] (3) τD ) QB/uvξNF [Baldyga and Bourne,2 eq 12.1, pp 730 and 763] (4) τS0 ) A(ΛC2/)1/3 ) 2(QB/πuvNF)1/3 [Baldyga et al.,2 eq E12.1e, p 764] (5) τSr ) 1.2(ΛCr2/) ) 1.2{[0.272(d/4)]2/}1/3 [Baldyga et al.,2 eq E12.1f, p 764] (6) The turbulent diffusivity (ξ) is needed to use eq 6. Hearn17 [p 48, Figure 7.25] developed correlations for turbulent diffusivities for four motionless mixers from measured residence time distribution data. At Re > ≈20 000, the intensity of dispersion parameter (ID ) ξ/ud), for the Kenics HEM mixer, is about 0.06. Equation 7 is a better fit of Hearn’s17 [p 48, Figure 7.25] curve of ID vs Re for the Kenics HEM mixer.
ID ) ξ/ud ) 0.06 + 0.01[(1 × 105/Re)0.5 - 1] (7) Hearn17 [Chapter 10, pp 221-234] and Baldyga and Bourne2 [example 12.4, pp 769-771] have covered the scale-up relationships for each of the four limiting time scales above. The pertinent scale-up procedures are covered below. For the limiting case of “micromixing”, must be held constant; thus
dnew ) (Qnew/Qold)3/7dold
[Hearn,17 eq 10.1, p 229;
Baldyga and Bourne,2 eq E12.4a, p 769] (8) Then, for a scale-up factor (Qnew/Qold) ) 100, dnew/dold ) (100)3/7 ) 2.68. The new mixer length to maintain equal residence time is given by
Lnew ) (Qnew/Qold)1/7Lold
[Hearn,17 eq 10.4, p 230;
Baldyga and Bourne,2 eq E12.4b, p 769] (9) Thus, Lnew/Lold ) (100)1/7 ) 1.93. Then (L/d)new ) (1.93/ 2.68)(L/d)old ) 0.72(L/d)old. According to these “micromixing” scaling recommendations, the number of mixing elements can be reduced on scale-up provided that, as preferred, one maintains the ratio of the mixing element length to the mixer diameter (Le/d) constant. The velocity is proportional to Q/d2; thus, unew/uold ) (Qnew/ Qold)(Qold/Qnew)2(3/7) ) (Qnew/Qold)1/7 and unew ) 1001/7uold ) 1.39uold. For “mesomixing” control, Baldyga and Bourne2 [p 771] say, “The scale-up in parts (b)-(d) are the same, because the mixing mechanism (inertial-convective) is unchanged”. Parts b and c to which they refer are the
mesomixing-controlled regimes mentioned above, and part d is the COV. For all of the regimes, except micromixing, Baldyga and Bourne2 show that
d ∝ Q1/3 and L ∝ Q1/3 [Baldyga and Bourne,2 eq E12.4b,c, p 770] (10a,b) Thus, L/d remains constant, and if Qnew/Qold ) 100, dnew/ dold ) (100)1/3 ) 4.64 and Lnew ) 4.64Lold. The velocity is proportional to Q/d2; thus, unew/uold ) (Qnew/Qold)(Qold/ Qnew)2(1/3) ) (Qnew/Qold)1/3 and unew ) 1001/3uold ) 4.64uold. The power dissipation is related to Q and d as follows: ∝ Q∆Pφ/d2L ∝ Qf(L/d)u2φ/d2L. Also, with a constant friction factor (f) and a constant fraction of energy dissipation converted to turbulence (φ), ∝ Q(Q/d2)2/d3 ∝ Q(Q/Q2/3)2/Q3/3 ∝ Q2/3. Thus, for scale-up of Qnew/Qold ) 100, new/old ) (Qnew/Qold)2/3 ) 1002/3 ) 21.6. The residence time in the mixer is related to Q and d as follows: τ ∝ L/u ∝ d/(Q/d2) ∝ Q1/3/(Q/Q2/3) ∝ 1; thus, the residence time must remain constant. An increase in by a factor of 21.6 is a high but necessary price to pay for scale-up with mesomixing control to maintain equal residence time and, thus, equal yield of the fast reaction for a pipeline mixer reactor. 2.3. Pressure Drop. Pressure drop correlations for the Kenics HEM mixers are presented in the Kenics (Chemineer) technical bulletin titled Kenics Static Mixers: KETK Series, pp 4-6 (May 1988), for an element length to diameter ratio (Le/d) of about 1.5. Myers et al.22 present correlations for the pressure drop for Le/d ) 1.0 and 1.5 and for a typical void fraction (φ) of approximately 0.9 (although the void fraction is not specified in ref 22). Hearn17 found for turbulent conditions that the Darcy friction factor was 1.9 based on the true void velocity (uv) in the mixer. The 1/4- and 1/2-in.-diameter mixers used in this investigation had an Le/d ) 1.5 but the 1/8-in.-diameter mixer had an Le/d ) 0.83. Pressure drop measurements by Taylor19 agreed well with the pressure drop correlation presented be Kenics and Myers et al.22 for NRe ∼ >2000, but the experimental measurements were up to 80% above the values obtained from the Kenics recommended correlation in the transitional Reynolds range (i.e., 100 < NRe < 2000). An improved correlation was developed for the transition regime, and it is documented by Taylor.19 The calculated pressure drop was verified by checks against experimental data. 2.4. Power Dissipation as Turbulence. Hearn17 (eqs 5.4-5.6, pp 70 and 71) gives the following equation:
) P/m ) Q∆Pφ/FVm ) [(πd2/4)u][(fL/d)Fu2/2]φ/F[Φ(πd2/4)L] ) fu3φ/2dΦ (11) 3. Experimental Apparatus and Procedure A schematic of the experimental setup is shown in Figure 2. The main and side feeds were fed to the static mixer using variable-speed gear pumps. The pump shaft speeds were monitored with electronic tachometers. The flow rates for both the main and side feeds were controlled manually by monitoring and adjusting a combination of (1) pump shaft speeds, (2) pump discharge pressures, and (3) mass flow rates. Both the main and side feed hold tanks were mounted on electronic scales. Electronic signals from the scales were sent to a digital computer, which computed the mass
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Figure 2. Process flow schematic of the experimental apparatus.
flow rate from incremental weight loss over short time increments. To obtain the best control of the side feed, a 1/16-in.-diameter capillary tube, of proper length, was installed between the pump and the HEM reactor. The 1/8-in.-diameter static mixer consisted of 12 1/ -in.-diameter polypropylene mixer elements with an 8 Le/d of 0.83. It was housed in a 1.5 in. × 1.5 in. × 2 in. Teflon block with two feed ports of 0.020-in. diameter drilled so that the feed entered at the midpoint of the third mixer element, normal to the side surfaces (i.e., not edge surfaces) of the element. The void fraction (Φ) was 0.678. Both of the 1/4- and 1/2-in.-diameter static mixers consisted of 12 elements, which were constructed from Teflon-coated steel. They had inside diameters of 0.277 and 0.507 in., two feed ports of 0.040 and 0.073 in., and void fractions of 0.787 and 0.839, respectively; the Le/d was 1.5 for both. The housings for the 1/4- and 1/2-in.diameter mixers were stainless steel tubes. To provide the side feed to these units, stainless steel distributor collars were constructed for each mixer. The compositions of the main and side feeds, along with the volumetric flow ratio, were as follows. Main feed: DMP mass fraction ) 0.0216 (200 gmol/m3), EtOH mass fraction ) 0.25, NaOH mass fraction ) 0.008 74 (210 gmol/m3), and NaCl mass fraction ) 0.006 08 (100 gmol/m3). Side feed: HCl mass fraction ) 0.073 (2000 gmol/m3), EtOH mass fraction ) 0.25, and volume flow ratio (main to side) ) 10:1. The analysis of the reaction products was performed using a Hewlett-Packard model 5890 series II gas chromatograph (GC) with flame ionization detectors. The GC column was a 30-m capillary, which contained a Supelco-Q plot and had a 0.53-mm internal diameter. The analysis used the ethanol solvent as an internal standard. This was possible because the ethanol concentration was 25 wt % in every stream. The acetone in the reactor effluent was used to close the mass balance. Additional details of the GC work are given by Taylor.19
Table 1. Xq and Reynolds Number Data for All Three Mixer Reactors 1/
8-in.-diameter
reactor (× on Figure 3)
1/
4-in.-diameter
reactor (O on Figure 3)
1/
2-in.-diameter
reactor (+ on Figure 3)
NRe
Xq
NRe
Xq
NRe
Xq
2500 1471 1100 550
0.041 0.063 0.09 0.145
8000 3762 1843 1000
0.11 0.17 0.30 0.39
17000 9900 4900 4500 2300
0.118 0.167 0.202 0.22 0.34
4. Correlation and Discussion of the Results 4.1. Analysis in Terms of the Reynolds Number, Specific Energy Dissipation, Residence Time in the Reactor, and COV. Prior work by Knight,13 Hearn,17 and Baldyga and Bourne2 has indicated that the following parameters are useful in developing a scale-up method: (1) equal power dissipation, which duplicates the time scale for micromixing; (2) equal residence time in the mixer, which duplicates time scales for mesomixing; (3) equal or lower COV, as indicated by Knight13 and Hearn;17 (4) Reynolds number. The Reynolds number is important because scaling from the laminar to the turbulent regime will perhaps invalidate any or all of the criteria listed above. However, in general, upon scale-up, the Reynolds number invariably increases, which should make any scale-up, using any of the criteria listed above, conservative. The Reynolds number of itself is never a proper scale-up criterion. 4.1.1. Effect of the Reynolds Number. The Reynolds number (NRe) was calculated on the basis of the superficial velocity, the pipe diameter, and the combined stream viscosity (2.5 cP). The densities were measured using a pycnometer and an analytical scale. The densities of both the side and main feed streams were the same, namely, 968 kg/m3. Table 1 presents the data of Xq and NRe for all three reactors, and Figure 3 presents graphically Xq vs NRe. The data for the 1/8-in.-diameter reactor bridges the transition regime for pipe flow. At equal NRe, there is a
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Figure 3. Yield of slow reaction (Xq) vs Reynolds number (Re) for all reactors. Legend: reactor (O); 1/2-in.-diameter reactor (+). Table 2. Xq and Energy Dissipation Rate (E) for All Three Mixer Reactors 1/
8-in.-diameter
reactor (× on Figure 4)
1/
4-in.-diameter
reactor (O on Figure 4)
1/
2-in.-diameter
reactor (+ on Figure 4)
(W/kg)
Xq
(W/kg)
Xq
(W/kg)
Xq
19800 4562 1854 251
0.041 0.063 0.09 0.145
4422 527 71 13.3
0.11 0.17 0.30 0.39
2746 594 82 64 9.7
0.118 0.167 0.202 0.22 0.34
large advantage for the 1/8-in.-diameter reactor because the residence time is much less and the power dissipation is much greater than those for either of the larger mixers, operating at the same NRe. Figure 3 also indicates that the slopes of the best-fit curves (not shown) through Xq vs NRe decrease as the mixer size increases. This is likely a result of the data bridging the transition regime of NRe because, as NRe increases in the transition regime, there should be an increase in the mixing rate, not just because more power is being dissipated but also because the flow is becoming more turbulent; i.e., more of the kinetic energy is being dissipated as turbulence. 4.1.2. Effect of the Energy Dissipation Rate. Table 2 presents the data of Xq and energy dissipation
1/
8-in.-diameter
reactor (×); 1/4-in.-diameter
() for all three reactors. Figure 4 presents graphically Xq vs . Note that, at the same , the 1/8-in.-diameter reactor gives consistently lower Xq values (i.e., better mixing) than either of the 1/4- or 1/2-in.-diameter reactors. This behavior is observed even though, as Figure 3 shows, NRe for the 1/8-in.-diameter unit is definitely in the lower portion of the transition flow regime, whereas NRe values for the 1/4- and 1/2-in.-diameter mixers are in the upper portion of the transition regime. One must be mindful of the fact that for the 1/8-in.diameter unit is considerably higher, for a given void velocity, than that for the larger units because the 1/8in.-diameter unit has a much tighter pitch on the helical elements (i.e., Le/d ) 0.83 vs 1.5 for the 1/4- and 1/2-in.diameter units, respectively) and the 1/8-in.-diameter unit also has a lower void volume fraction (Φ ) 0.687), which also increases the pressure drop for a given uv. The effects of the lower Le/d and Φ are included in Taylor’s19 pressure drop correlation; however, there are important facts to keep in mind when comparing the data from the 1/8-in.-diameter mixer with the data from the larger units, namely, the geometry of the 1/8-in.diameter unit is different (it has a lower void fraction, and it has a tighter twist on the elements, giving a lower Le/d). Thus, any difference shown by any correlating
Figure 4. Yield of slow (hydrolysis) reaction (Xq) vs energy dissipation rate () for all reactors. Legend: 1/ -in.-diameter reactor (O); 1/ -in.-diameter reactor (+). 4 2
1/
8-in.-diameter
reactor (×);
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Table 3. Xq and Residence Time Data for All Three Mixer Reactors 1/
8-in.-diameter
1/
4-in.-diameter
1/
2-in.-diameter
reactor (× on Figure 4)
reactor (O on Figure 4)
reactor (+ on Figure 4)
τ (s)
Xq
τ (s)
Xq
τ (s)
Xq
0.011 0.018 0.024 0.048
0.041 0.063 0.090 0.145
0.034 0.072 0.147 0.270
0.11 0.17 0.30 0.39
0.057 0.098 0.197 0.216 0.420
0.118 0.167 0.202 0.220 0.340
technique may be partly due to the different geometry and not to other independent parameters. The 1/8-in.-diameter mixer elements were selected because that is what is commercially available in molded polypropylene and the geometries of the 1/4- and 1/ -in.-diameter mixer elements were also what are 2 available commercially in Teflon-coated stainless steel. All three of the mixer units were previously used by Dow Corning Corp., who provided financial support for the project. The Teflon and polypropylene were needed to handle the corrosive reagents used at Dow Corning Corp. A comparison of Figures 3 and 4 clearly shows that equal is superior to equal NRe as a scale-up criterion. However, on the basis of the data of this study, a conservative scale-up method cannot be based on the energy dissipation rate because, in Figure 4, all of the 1/ -in.-diameter data and some of the 1/ -in.-diameter 4 2 data lie above the 1/8-in.-diameter data, which is nonconservative on the normal scale-up basis of meeting or exceeding the yield of the fast reaction (i.e., minimizing Xq) upon scale-up from laboratory or pilot plant to plant. 4.1.3. Residence Time in the Mixer. Table 3 presents the Xq and residence time (τ) data for all three mixers. Figure 5 presents graphically Xq vs τ. The data generally correlate well, with the exception being the higher residence time of the 1/2-in.-diameter data. Note from Table 1 and Figure 3 that the highest NRe in the 1/ -in.-diameter reactor is 17 000, whereas N 2 Re in the 1/ -in.-diameter reactor varied from 550 to 2500; thus, 8 it is expected that, at the same τ, the 1/2-in.-diameter reactor would outperform the 1/8-in.-diameter reactor
because the flow is considerably more turbulent. On a conservative basis, equal τ, which keeps the actual time scales constant on scale-up, seems to be the most prudent scale-up criterion because it is expected to be conservative, and Figure 4 shows that scale-up at equal is not conservative. 4.1.4. COV. The COV (i.e., the standard deviation/ mean) is the most common means of characterizing the blending performance of static mixers. Myers et al.22 have presented correlations for determining COV for the Kenics HEM mixers. The role of COV in scale-up of motionless pipeline mixers for handling C/C fast reactions has been considered by Hearn17 and Baldyga and Bourne.2 Hearn17 [p 231] says, “The COV correlations ... showed that the macromixing length required to achieve a given COV ... was insensitive to Re for the HEV and Kenics mixers for fully turbulent conditions. However, the macromixing time is inversely proportional to the pipe velocity. Therefore, although the mixing length remains approximately constant, on increasing pipe velocity, the time for mixing will be shorter for a high velocity. For reacting systems where macromixing is important, the COV and residence time must both be kept constant on scale-up.” Baldayga and Bourne2 [p 771] say, “The COV is proportional to the standard deviation of the radial concentration determined at some axial position. It has been observed to be independent of flow rate in a given mixer once turbulence was fully developed but to decay as a function of L/d which is proportional to the number of elements. Turbulent dispersion is the relevant mixing mechanism and the results in part (b) [Note: (b) was for turbulent dispersion control] are again applicable.” Thus, any influence of COV on the reactor performance is covered by the scaleup criteria of the mixing time scales. 5. Analysis Considering Micro- and Mesomixing Time Scales The time scales (or time constants) identified by Baldyga and Bourne,2 presented here as eqs 3-6, are used here for determining the appropriate time scales. 5.1. Evaluation of the Data in Light of Microand Mesomixing Time Scales. Tables 4-6 present the data for the various time scales along with the yield,
Figure 5. Yield of slow reaction (Xq) vs residence time (τ) for all reactors. Legend: 1/8-in.-diameter reactor (×); 1/4-in.-diameter reactor (O); 1/2-in.-diameter reactor (+).
Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 6101 Table 4. Yield, Reynolds Number, Energy Dissipation, and Mixing Time Scales (τE, τD, τS0, and τSr) for Experiments in the 1/8-in.-Diameter Reactor Xq
NRe
0.145 550 0.090 1100 0.063 1471 0.041 2500
(W/kg)
τE (s)
τD (s)
τS0 (s)
τSr (s)
251 1854 4562 19800
0.001 75 0.000 65 0.000 41 0.000 20
0.000 93 0.000 59 0.000 48 0.000 34
0.001 35 0.000 69 0.000 51 0.000 32
0.000 68 0.000 35 0.000 26 0.000 16
Table 5. Yield, Reynolds Number, Energy Dissipation, and Mixing Time Scales (τE, τD, τS0, and τSr) for Experiments in the 1/4-in.-Diameter Reactor Xq
NRe
(W/kg)
τE (s)
τD (s)
τS0 (s)
τSr (s)
0.11 0.17 0.30 0.39
1000 1843 3762 8000
13.3 71 527 4422
0.007 63 0.003 31 0.001 21 0.000 42
0.003 58 0.002 36 0.001 41 0.000 79
0.006 44 0.003 69 0.001 89 0.000 93
0.003 10 0.001 78 0.000 91 0.000 45
Table 6. Yield, Reynolds Number, Energy Dissipation, and Mixing Time Scales (τE, τD, τS0, and τSr) for Experiments in the 1/2-in.-Diameter Reactor Xq
NRe
0.11 2300 0.27 4500 0.20 4900 0.0.22 9900 0.34 17000
(W/kg)
τE (s)
τD (s)
τS0 (s)
τSr (s)
9.7 64 82 594 2746
0.008 92 0.003 47 0.003 08 0.001 41 0.000 53
0.007 20 0.004 39 0.004 12 0.002 37 0.001 52
0.010 92 0.005 82 0.005 37 0.002 77 0.001 66
0.005 15 0.002 74 0.002 53 0.001 31 0.000 78
Reynolds number, energy dissipation, and residence time for the three reactors. For the 1/8-in.-diameter reactor, micromixing controls at the lowest NRe ) 550, whereas at the higher NRe, mesomixing of turbulent dispersion and turbulent disintegration of the entering feed are the controlling mechanisms. For the 1/4-in.-diameter reactor, micromixing controls for the lowest Re ) 1000, but just barely. At all higher NRe, mesomixing of turbulent dispersion and turbulent disintegration of the entering feed are the controlling mechanisms. For the 1/2-in.-diameter reactor, mesomixing of turbulent dispersion of the entering feed is the controlling mechanism. From Figure 4, the yield data for all three reactors are much closer together at the lowest , whereas the yield data are significantly higher for the two larger reactors than those for the 1/8-in.-diameter reactor at the highest . This behavior is consistent with the controlling time scales because micromixing controls at low and mesomixing controls at high . The message from Figure 5 is the same as the message from Figure 4: At low τ (i.e., high ), the yield data for all reactors converge, indicating that equal τ is the proper scaling parameter where mesomixing controls, and at high τ (i.e., low ), scale-up at equal τ will result in a better performance for the plant reactor, indicating that scaling at equal is acceptable (but not necessarily conservative) when micromixing controls. 6. Concluding Remarks The fourth Bourne reaction is sufficiently fast to allow the investigation of static pipeline mixers as reactors for fast C/C reactions. The required chemicals are relatively nontoxic and readily available, and the sample analysis to determine acetone and methanol in an aqueous solution is straightforward. Micromixing was the controlling mechanism at low flow rates and low energy dissipation, but mesomixing
was the controlling mechanism at high flow rates and high energy dissipation. With micromixing control, equal power dissipation is an acceptable (but not necessarily conservative) scaleup criterion. With mesomixing control, equal residence time is the proper (and conservative) scale-up criterion. Equal residence time is always a conservative scaleup criterion. For example, the experimental results from Figure 5, at τ ) 0.05 s, where Xq,1/8 ) 0.11 and Xq,1/2 ) 0.15, indicates an improvement (i.e., a decrease) in the yield of the slow reaction of 100[(0.15 - 0.11)/0.15] ) 26% when using equal residence time as a scaling criterion. From Figure 4, scale-up of the 1/8-in.-diameter reactor from the lowest flow condition at ) 250 W/kg to that of the 1/2-in.-diameter reactor, also at ) 250 W/kg, both yields of the slow reaction are about the same at Xq,1/8 ) Xq,1/2 ) 0.18. This study should help future studies to better define the proper scale-up criteria over a wider range of parameters. The 1/8-in.-diameter mixer had a different geometry than the 1/4- and 1/2-in.-diameter units; the 1/8in.-diameter unit had shorter elements (Le/d ) 0.83 vs 1.5) and a lower void fraction (φ ) 0.678 vs 0.787 and 0.839). In future studies, the geometries of the mixing elements should be the same for initial testing. Any effect of the element geometry must be tested separately from the effect(s) of other variables. Care should be exercised to use side feed flow distributors, which give radial discharge through the side ports, and one should be extra careful that the side ports are properly aligned with the mixing elements, both axially and circumferentially. Nomenclature A ) constant in eq 5 COV ) coefficient of variation ()standard deviation/mean) d ) inside diameter of mixer housing, m E ) engulfment rate coefficient, 1/s f ) friction factor for the mixer ID ) dimensionless intensity of the dispersion parameter (ξ/ud) L ) overall length of the mixer, m Le ) length of a mixing element length, m m ) mass of the fluid within the mixer, kg NF ) number of side feed ports NRe ) mixer Reynolds number ()udF/µ) P ) power dissipated within the mixer, W Q ) volumetric flow rate, m3/s QA ) volumetric flow rate of the main stream, m3/s QB ) volumetric flow rate of the side stream, m3/s Qratio ) volumetric flow ratio (main/side ) QA/QB) u ) superficial velocity through the mixer, m/s uv ) fluid velocity through the void volume of the mixer, m/s ()u/Φ) Xq ) fractional conversion of DMP to acetone and methanol Greek Symbols ∆P ) pressure drop through the mixer, Pa ) turbulent energy dissipation, W/kg Φ ) void fraction in the mixer φ ) efficiency of energy dissipation converted to turbulence, )0.65 for the Kenics HEM mixer ΛC ) integral concentration scale, m µ ) dynamic fluid viscosity, cP or mPa‚s ν ) ()µ/F) kinematic fluid viscosity, m2/s F ) fluid density, kg/m3 τ ) residence time in the mixer, s
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τD ) mesomixing time scale for turbulent dispersion, eq 4, s τE ) time scale for micromixing, eq 3, s τS0 ) mesomixing time scale: turbulent disintegration of the entering feed, s τSr ) mesomixing time scale: turbulent disintegration of the relaxed feed, s ξ ) turbulent diffusivity, m2/s
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(12) Yu, S. Micromixing and Parallel Reactions. Ph.D. Dissertation, Swiss Federal Institute of Technology, Zurich, Switzerland, 1993. (13) Knight, C. S. Experimental Investigation of the Effects of a Recycle Loop/Static Mixer/Agitated Vessel System on Fast, Competitive-Parallel Reactions. M.S. Thesis, University of Arkansas, Fayetteville, AR, 1994. (14) Knight, C. S.; Penney, W. R.; Fasano, J. B. Experimental Investigation of Effects of a Recycle Loop/Static Mixer/Agitated Vessel System on Fast, Competitive-Parallel Reactions. Paper written and presented at the AIChE Annual Meeting, Miami Beach, FL, Nov 1995; available from WRP. (15) Penney, W. R.; Knight, C. S.; Fasano, J. B. Fast Competitive Reactions in Agitated Vessels: Status of Scale-up and Design Procedures Using In-Tank Agitators and Static Mixer in Recycle Loops. Paper written and presented at the 15th NAMF Mixing Conference, Banff, Alberta, Canada, June 1996; available from WRP. (16) Bourne, J. R.; Lenzner, J.; Petrozzi, S. Micromixing in Static Mixers: An Experimental Study. Ind. Eng. Chem. Res. 1992, 31 (No. 4), 1216-1222. (17) Hearn, S. Turbulent Mixing Mechanisms in Motionless Mixers. Ph.D. Dissertation, University of Birmingham, Birmingham, U.K., 1995. (18) Baldyga, J.; Bourne, J. R.; Hearn, S. J. Interaction Between Chemical Reactions and Mixing on Various Scales. Chem. Eng. Sci. 1997, 52 (No. 4), 457-466. (19) Taylor, R. A. Scale-up Methods for Fast Competitive Chemical Reactions in Pipeline Mixers. M.S. Thesis, University of Arkansas, Fayetteville, AR, 1998. (20) Walker, B. M. Einfluss der Temperatur-Segregation auf die Selektivitat rasch ablaufender Reaktionen. Ph.D. Dissertation, Swiss Federal Institute of Technology, Zurich, Switzerland, 1996. (21) Baldaga, J.; Bourne, J. R.; Walker, B. Non-isothermal Micromixing in Liquids: Theory and Experiment. Can. J. Chem. Eng. 1998, 76, 641-649. (22) Myers, K. J.; Bakker, A.; Ryan, D. Avoid Agitation by Selecting Static Mixers. Chem. Eng. Prog. 1997, 93 (No. 6), June, 28-38.
Received for review September 1, 2004 Revised manuscript received January 10, 2005 Accepted January 11, 2005 IE040237U
